The "Administrative password" that it's asking for is the
root password, which is not set by default. You have two solutions:
You can set the
root password, and then use that password to install your driver. However, it is best practice to not set a
root password unless you really need to, because that opens up lots of other doors that may lead to accidental deletion of stuff and all that, so you should go with solution number 2.
Open a terminal by Ctrl+Alt+T, and do
sudo /path/to/driver. So, let's say you downloaded the driver in your Downloads folder, you would do
sudo ~/Downloads/filename. The password that it will ask for is your username's password.
On a side note though, what is the extension of this printer driver? Because I wanted to see for myself, so I visited the Lexmark Support & Downloads page for that printer, chose OS = Linux, version = Ubuntu 12.04, and all the files are in .deb format, so these shouldn't ask you to "Run in terminal" when you double click them.